The \(\pi \) -values to improve \(\alpha \) -measure.
In order to further enhance the advantage of the candidate set, LKH uses a method to maximize the lower bound (i.e., the length of the minimum 1-tree) of the optimal TSP solution by adding penalties (\(\pi \) -values) ([1]} [1]}). Concretely, a \(\pi \) -value computed using a sub-gradient optimization method ([3]} [3]}) is added to each node as penalties when calculating the distance between two nodes:
\(C(i,j)=d(i,j)+\pi _i+\pi _j,\) 
